Quantum Interference and Coherence: Theory and Experiments PDF
Preview Quantum Interference and Coherence: Theory and Experiments
Springer Series in 100 OPTICAL SCIENCES Founded by H.K.V. Lotsch Editor-in-Chief: W.T. Rhodes, Atlanta Editorial Board: T. Asakura, Sapporo K.-H. Brenner, Mannheim T.W. Ha¨nsch, Garching T. Kamiya, Tokyo F. Krausz, Vienna and Garching B. Monemar, Linko¨ping H. Venghaus, Berlin H. Weber, Berlin H. Weinfurter, Munich Springer Series in OPTICAL SCIENCES TheSpringerSeriesinOpticalSciences,undertheleadershipofEditor-in-ChiefWilliamT.Rhodes, Georgia Institute of Technology, USA, and Georgia Tech Lorraine, France, provides an expanding selection of research monographs in all major areas of optics: lasers and quantum optics, ultrafast phenomena,optical spectroscopytechniques,optoelectronics, quantuminformation, informationop- tics,appliedlasertechnology,industrialapplications,andothertopicsofcontemporaryinterest. Withthisbroadcoverageoftopics,theseriesisofusetoallresearchscientistsandengineerswho needup-to-datereferencebooks. Theeditorsencourageprospectiveauthorstocorrespondwiththeminadvanceofsubmittingamanu- script.SubmissionofmanuscriptsshouldbemadetotheEditor-in-ChieforoneoftheEditors. Editor-in-Chief WilliamT.Rhodes FerencKrausz GeorgiaInstituteofTechnology Max-Planck-Institutfu¨rQuantenoptik SchoolofElectricalandComputerEngineering Hans-Kopfermann-Strasse1 Atlanta,GA30332-0250,USA 85748Garching,Germany E-mail:[email protected] and ViennaUniversityofTechnology PhotonicsInstitute EditorialBoard Gusshausstrasse27/387 1040Wien,Austria ToshimitsuAsakura E-mail:[email protected] Hokkai-GakuenUniversity FacultyofEngineering BoMonemar 1-I,Minami-26,Nishi11,Chuo-ku DepartmentofPhysics Sapporo,Hokkaido064-0926,Japan andMeasurementTechnology E-mail:[email protected] MaterialsScienceDivision Linko¨pingUniversity 58183Linko¨ping,Sweden Karl-HeinzBrenner E-mail:[email protected] ChairofOptoelectronics UniversityofMannheim HerbertVenghaus InstituteofComputerEngineering Heinrich-Hertz-Institut B6,26 fu¨rNachrichtentechnikBerlinGmbH 68131Mannheim,Germany Einsteinufer37 E-mail:[email protected] 10587Berlin,Germany E-mail:[email protected] TheodorW.Ha¨nsch HorstWeber Max-Planck-Institutfu¨rQuantenoptik Hans-Kopfermann-StrasseI TechnischeUniversita¨tBerlin 85748Garching,Germany OptischesInstitut E-mail:[email protected] Strassedes17.Jun135 10623Berlin,Germany E-mail:[email protected] TakeshiKamiya MinistryofEducation,Culture,Sports HaraldWeinfurter ScienceandTechnology Ludwig-Maximilians-Universita¨tMu¨nchen NationalInstitutionforAcademicDegrees SektionPhysik 3-29-1Otsuka,Bunkyo-ku Schellingstrasse4/III Tokyo112-0012,Japan 80799Mu¨nchen,Germany E-mail:[email protected] E-mail:[email protected] Zbigniew Ficek Stuart Swain Quantum Interference and Coherence Theory and Experiments With 179 Figures ZbigniewFicek StuartSwain DepartmentofPhysics SchoolofMathematicsandPhysics TheUniversityofQueensland Queen’sUniversityBelfast Brisbane,QLD4072 Belfast,BT71NN Australia UK [email protected] [email protected] LibraryofCongressCataloging-in-PublicationData Ficek,Zbigniew. Quantuminterferenceandcoherence:theoryandexperiments/ZbigniewFicekand StuartSwain. p. cm.—(Springerseriesinopticalsciences,ISSN0342-4111) Includesbibliographicalreferencesandindex. ISBN0-387-22965-5(alk.paper) 1.Quantuminterference. 2.Coherentstates. 3.Interference(Light) 4.Coherence (Nuclearphysics) 5.Quantumtheory. I.Swain,Stuart. II.Title. III.Series. QC174.17.Q33F53 2004 535′.15—dc22 2004051296 ISBN0-387-22965-5 Printedonacid-freepaper. ©2005SpringerScience+BusinessMedia,Inc. Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permissionof thepublisher (SpringerScience+Business Media,Inc., 233Spring Street,New York, NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Usein connectionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoft- ware,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,evenifthey arenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyare subjecttoproprietaryrights. PrintedintheUnitedStatesofAmerica. (BS/SBA) 9 8 7 6 5 4 3 2 1 SPIN10949054 springeronline.com Dedicated to our wives Agnieszka Lichan´ska and Gisela Ilse in appreciation of their help, patience and understanding Preface The field that encompasses the term “quantum interference” combines a number of separate concepts, and has a variety of manifestations in dif- ferent areas of physics. In the sense considered here, quantum interference is concerned with coherence and correlation phenomena in radiation fields and between their sources. It is intimately connected with the phenomenon of non-separability (or entanglement) in quantum mechanics. On account of this, it is obvious that quantum interference may be regarded as a compo- nent of quantum information theory, which investigates the ability of the electromagnetic field to transfer information between correlated (entangled) systems. Since it is important to transfer information with the minimum of corruption, the theory of quantum interference is naturally related to the theory of quantum fluctuations and decoherence. Since the early days of quantum mechanics, interference has been de- scribed as the real quantum mystery. Feynman, in his famous introduction to the lectures on the single particle superposition principle, referred in the following way to the phenomenon of interference: “it has in it the heart of quantummechanics”,anditisreally‘theonlymystery’ofquantummechan- ics.Withthedevelopmentofexperimentaltechniques,ithasbeenpossibleto carryoutmanyoftheearlyGedankenexperimentsthatplayedanimportant role in developing our understanding of the fundamentals of quantum inter- ferenceandentanglement.Despiteitslonghistory,quantuminterferencestill challenges our understanding, and continues to excite our imagination. Quantum interference arises in some form or other in almost all the phe- nomena of quantum mechanics and its applications. Obviously, we have to be very selective in the topics we discuss here, and many important aspects are dealt with only briefly, or not at all. In writing the book our intention has been to concentrate on a systematic and consistent exposition of co- herence and quantum interference phenomena in optical fields and atomic systems and to discuss the details of the most recent theoretical and ex- perimental work in the field. We begin in Chap. 1 by discussing the basic principlesofclassicalandquantuminterferenceandsummarizingsomequite elementary concepts and definitions that are frequently used in the analysis of interference phenomena. The most important first- and second-order co- herence effects are discussed including the welcher-weg problem, two-photon VIII Preface nonclassical interference, interferometric interaction-free measurements, and quantum lithography. We also discuss important experiments that confirm these basic interference predictions. The mathematical formalism of quantum interference in atomic systems is developed in Chap. 2 for multi-level and multi-atom systems in free space andcavityenvironments.Forourpurposes,themasterequationofanatomic system is derived in the Born−Markov and rotating-wave approximations. The relation of the source field operators to the atomic dipole operators and retardation effects are then discussed. In this way the correlation functions of the electric field and their relationship to the atomic dipole operators are developed as a basic formulation. The concept of superposition states is then introduced in Chap. 3 and applied to three-level systems in Vee and Lambda configurations. The concept of multi-atom entangled states is also introduced so that one can see the relation between quantum interference effects in multi-level and multi-atom systems. A full description of the quan- tum beats phenomenon and its relation to quantum interference phenomena is also included. Chapter 4 discusses quantum interference effects induced by spontaneous emission and the experimental evidence of spontaneously induced quantum interference effects in a molecular multi-level system. This chapter includes a discussion of decoherence free subspaces and the role of decoherence in the formation of entanglement. A section on the effect of cavity and photonic bandgap materials on spontaneous emission from an atomic system is in- cluded here because these are examples of other practical systems to control and suppress spontaneous emission. The subject of coherence effects in multi-level systems is treated in Chap. 5. The theory of two major quantum interference effects − coherent population trapping and electromagnetically induced transparency in simple three-level systems − are explored and described in terms of the density ma- trixelementsofthesesystems.Theseprocessesdependonthecreationofco- herentsuperpositionsofatomicstateswithaccompanyinglossofabsorption. The chapter includes a general treatment of the spatial propagation of elec- tromagnetic fields in optically dense media, and the absorption properties of coherentlypreparedatomicsystems.Thischapteralsodiscussesapplications of coherently prepared systems in the enhancement of optical nonlinearities in electromagnetically induced transparency. Material on the implementation of quantum interference is included in Chap.6.Thischapteralsodiscussesthephasecontrolofquantuminterference and extremely large values (superbunching) of the second-order correlation functions. Methods for producing quantum interference effects in three-level systems with perpendicular transition dipole moments is considered to show howonecangetaroundthewell-knowndifficultyoffindingatomicormolec- ularsystemswithparalleltransitiondipolemoments.Thischapterconcludes Preface IX with a fairly detailed description of Fano profiles, laser-induced continuum structures and population trapping in photonic bandgap materials. In Chap. 7 the theory of subluminal and superluminal propagation of a weak electromagnetic field in coherently prepared media is formulated and accompanied with many examples of the experimental observation of slow and fast light, and the storage of photons. The concept of polaritons is then introduced in terms of atomic and field operators. The subject of quantum interference in a superposition of field states is considered in Chap. 8. The phase space formalism is described and quan- tum interference effects in phase space for several field states are discussed. Examples of the experimental reconstruction of Wigner functions and of the production of single-photon states are also included. The final chapter discusses quantum interference effects with cold atoms. This includes the subjects of diffraction of cold atoms, interference of two Bose−Einstein condensates, collapses and revivals of an atomic interference pattern and interference experiments in coherent atom optics. Since this book is based to a large extent on the combined work of many earlier contributors to the field of quantum interference, it is impossible to acknowledge our debts on an individual basis. We should, however, like to express our thanks to Peng Zhou who, during his stay at The Queen’s Uni- versity of Belfast, carried out some of the work on control of decoherence and field induced quantum interference presented in Chaps. 4 and 6. We are greatly appreciative of the help and suggestions received from many col- leagues, including Ryszard Tana´s, Helen Freedhoff, Peter Drummond, Bryan Dalton, Shi-Yao Zhu, Christoph Keitel, Josip Seke, Gerhard Adam, Andrey Soldatov,JoergEvers,TerryRudolphandUzmaAkram.Wearealsograteful to Alexander Akulshin, Immanuel Bloch, Dmitry Budker, Milena D’Angelo, Juergen Eschner, Edward Fry, Christian Hettich, Alexander Lvovsky, Steven Rolston, and Lorenz Windholz for sending us originals of the reproduced figures of their experimental results. Brisbane, Belfast, Zbigniew Ficek March 2004 Stuart Swain Contents 1 Classical and Quantum Interference and Coherence............................................ 1 1.1 Classical Interference and Optical Interferometers .......... 2 1.1.1 Young’s Double Slit Interferometer ................. 2 1.1.2 First-Order Coherence ............................ 4 1.1.3 Welcher Weg Problem ............................ 7 1.1.4 Experimental Tests of the Welcher Weg Problem ..... 11 1.1.5 Second-Order Coherence .......................... 15 1.1.6 Hanbury-Brown and Twiss Interferometer .......... 17 1.1.7 Mach−Zehnder Interferometer ..................... 19 1.2 Principles of Quantum Interference ....................... 20 1.2.1 Two-Photon Nonclassical Interference............... 21 1.2.2 The Hong−Ou−Mandel Interferometer ............. 25 1.3 Quantum Erasure ...................................... 28 1.4 Quantum Nonlocality ................................... 30 1.5 Interferometric Interaction-Free Measurements ............. 32 1.5.1 Negative-Result Measurements ..................... 33 1.5.2 Schemes of Interaction-Free Measurements........... 34 1.6 Quantum Interferometric Lithography .................... 38 1.7 Three-Photon Interference .............................. 42 1.7.1 Three-Photon Classical Interference................. 43 1.7.2 Three-Photon Nonclassical Interference ............. 44 2 Quantum Interference in Atomic Systems: Mathematical Formalism ................................................ 47 2.1 Master Equation of a Multi-Dipole System................. 48 2.1.1 Master Equation of a Single Multi-Level Atom ....... 48 2.1.2 Master Equation of a Multi-Atom System ........... 67 2.2 Correlation Functions of Atomic Operators ................ 74 2.2.1 Correlation Functions for a Multi-Level Atom........ 74 2.2.2 Correlation Functions for a Multi-Atom System ...... 80 2.2.3 Spectral Expressions.............................. 82 XII Contents 3 Superposition States and Modification of Spontaneous Emission Rates ........................................... 85 3.1 Superposition States in a Multi-Level System .............. 85 3.1.1 Superpositions Induced by Spontaneous Emission..... 87 3.2 Multi-Atom Superposition (Entangled) States.............. 91 3.2.1 Entanglement.................................... 91 3.2.2 Two Interacting Atoms ........................... 93 3.2.3 Entangled States of Two Identical Atoms............ 94 3.2.4 Entangled States of Two Nonidentical Atoms ........ 96 3.3 Experimental Evidence of the Collective Damping and Frequency Shift ........................................ 104 3.4 General Criteria for Interference in Two-Atom Systems...... 110 3.4.1 Interference Pattern with Two Atoms ............... 111 3.4.2 Experimental Observation of the Interference Pattern in a Two-Atom System............................ 113 3.5 Quantum Beats ........................................ 115 3.5.1 Theory of Quantum Beats in Multi-Level Systems .... 116 3.5.2 Quantum Beats in the Radiation Intensity from a Multi-Level Atom ................................ 120 3.5.3 Quantum Beats in the Radiation Intensity from Two Nonidentical Atoms............................... 126 3.5.4 Experimental Observation of Quantum Beats in a Type I System ................................... 129 3.5.5 Quantum Beats in the Intensity−Intensity Correlations ..................................... 131 3.6 Interference Pattern with a Dark Center................... 135 4 Quantum Interference as a Control of Decoherence ....... 139 4.1 Modified Spontaneous Emission .......................... 139 4.1.1 Effect of Environment on Spontaneous Emission...... 140 4.1.2 Modification by a Moderate Q Cavity............... 142 4.1.3 Modification by Photonic Crystals.................. 145 4.2 Quantum Interference in Vee Systems ..................... 146 4.2.1 Population Trapping and Dark States............... 148 4.2.2 Probing Quantum Interference in a Vee System ...... 150 4.3 Spectral Control of Spontaneous Emission ................. 156 4.4 Experimental Evidence of Quantum Interference............ 162 4.4.1 Energy Levels of the Molecular System.............. 162 4.4.2 Master Equation of the System..................... 163 4.4.3 Two-Photon Excitation ........................... 164 4.4.4 One- and Two-Photon Excitations.................. 166 4.5 Decoherence Free Subspaces ............................. 169 4.5.1 A Simple Example of a Decoherence Free Subspace ... 169 4.5.2 Experimental Verification of Decoherence Free Subspaces ....................................... 171
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